International Finance: Putting Theory Into Practice

5.2. USING FORWARD CONTRACTS (1): ARBITRAGE 177

5.2.1 Synthetic Forward Rates

It should not come as a surprise to you that, in the presence of spreads, the synthetic
forward rates are the Worst Possible Combinations of the basic perfect-markets for-
mula. We can immediately see this when we do the two trips on the diagram in
Figure 5.3. In the diagram we started from the figures familiar from last chapter,
but now setting the bids slightly below and the asks slightly above the old levels.
Let’s figure out the synthetic rates:

• Synthetic bid: the synthetic-sale trip isFCT→FCt→HCt→HCT, and it
  yields

HCT = FCT×

1

1. 101

× 99. 99 × 1. 209 , (5.1)

⇒syntheticFt,Tbid =

HCT

FCT

= 99. 99

1. 209

1. 101

= 109. 798. (5.2)

• Synthetic ask: the synthetic-purchase trip isHCT →HCt→FCt→FCT,
  and it yields

FCT = HCT×

1

1. 211

×

1

100. 01

× 1. 099 , (5.3)

⇒syntheticFt,Task =

HCT

FCT

= 100. 01

1. 211

1. 099

= 110. 202. (5.4)

We see that, in computing the synthetic bid rate, we retain the basic CIP formula
but add thebidoraskqualifiers that generate the lowest possible combination:bid
× bid / ask. Likewise, in computing the synthetic ask rate we pick the highest
possible combination:ask×ask / bid. In short,

synthetic [Ft,Tbid, Ft,Task] =

[

Sbidt

1 +rt,Tbid

1 +rt,T∗ask

, Saskt

1 +raskt,T

1 +r∗t,Tbid

]

(5.5)

5.2.2 Implications of Arbitrage and Shopping-around

In Figure 5.4, we illustrate the by now familiar implications of the arbitrage and
shopping-around mechanisms:

1. Arbitrage ensures that the synthetic and actual quotes can never be so far
   apart that there is empty space between them. Thus, given the synthetic
   quotes 109.8-110.2, we can rule out Case 1: we would have been able to buy
   directly at 109.7 and sell synthetically at 109.8. Likewise, situations like Case
   2 should vanish immediately (if they occur at all): we would have been able
   to buy synthetically at 110.2 and sell at 110.3 in the direct market.